Move from the smallest to the largest

Techniques for speeding up merging of multiple sets

• When merging something of size X into something of size Y, it is assumed to hang O(X). python

for x in X:
    Y.add(x)

• Repeat the merging of the elements of a set of size N into a set of size 1 with only the elements of each

  • In a naive way, O(N) merges N-1 times, so O(N^2)
  • 

• If we decide to merge the smaller one into the larger one, this becomes O(NlogN)

  • Discussion:.
    • Let X be the smaller one.
    • Each time you merge, the size of the set will be 2X or larger.
    • If we look at an element x, it is at most ${\log }_{2}N$ times the smaller of the two.
    • 

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